Question

The circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $(x_i,y_i)$, for $i=1,2,3$ and $4$, then $y_1+y_2+y_3+y_4$ equals

• A. $0$
• B. $c$
• C. $a$
• D. $c^2$

Answer 1

The correct option is A

$0$

Explanation of correct option:

Given information are as follows:

Equation of circle is $x^2+y^2=a^2$.

Equation of hyperbola is $xy=c^2$.

$\therefore x=\frac{c^2}{y}$.

Substituting value of $x$ in $x^2+y^2=a^2$.

$$\begin{gathered} \Rightarrow y^2+\frac{c^4}{y^2}=a^2 \\ \Rightarrow y^4+c^4=a^2{y}^2 \\ \Rightarrow y^4-a^2{y}^2+c^4=0 \end{gathered}$$

The roots of this equation is $y_1,y_2,y_3,y_4$.

And the sum of the roots is $0$.

Therefore, the correct answer is Option (A).